Abstract

The sectional curvature of the volume preserving diffeomorphism group of a
Riemannian manifold $ M$ can give information about the stability of
inviscid, incompressible fluid flows on $ M$. We demonstrate that the
submanifold of the volumorphism group of the solid flat torus generated by
axisymmetric fluid flows with swirl, denoted by $ \DiffmuE(M)$, has positive
sectional curvature in every section containing the field $ X = u(r)\partial_\theta$ iff
$ \partial_r(ru^2)>0$. This is in sharp contrast to the situation on $ \Diffmu(M)$, where
only Killing fields $ X$ have nonnegative sectional curvature in all
sections containing it. We also show that this criterion guarantees the
existence of conjugate points on $ \DiffmuE(M)$ along the geodesic defined by
$ X$.

Keywords

1.
Introduction

Let $ (M,g)$ be a Riemannian manifold of dimension at least two
with Riemannian volume form $ \mu$. The configuration space for
inviscid, incompressible fluid flows on $ M$ is the collection of smooth
volume-preserving diffeomorphisms (volumorphisms) of $ M$, denoted
by $ \mathcal{D}_\mu(M)$. [Arnold2014] showed in 1966 that flows obeying the
Euler equations for inviscid, incompressible fluid flow can formally This was
proved rigorously by [Ebin and Marsden1970], by working in the context
of Sobolev $ H^s$ diffeomorphisms for $ s>\tfrac{1}{2}\dim{M}+1$. Here for simplicity
we will work in the context of smooth diffeomorphisms since the curvature
formulas are the same either way. ×1 be realized as geodesics on
$ \mathcal{D}_\mu(M)$. Using this framework, questions of fluid mechanics can be
re-phrased in terms of the Riemannian geometry of $ \Diffmu(M)$. An overview
of this is given in [Arnold and Khesin1998] or more recently in [Khesin et al.2013]. Of particular interest is the
sectional curvature of $ \Diffmu(M)$. As in finite dimensional geometry, given
two geodesics with varying initial velocities in a region of strictly positive
(resp. negative) sectional curvature, the two geodesics will converge (resp.
diverge) via the Rauch Comparison theorem. In terms of fluid mechanics, this
corresponds to the Lagrangian stability (resp. instability) of the associated
fluid flows.

Arnold showed that the sectional curvature $ K(X,Y)$ of the plane
in $ T_{\id}\Diffmu(M)$ spanned by $ X$ and $ Y$ is often negative
but occasionally positive. [Rouchon1992] sharpened this to show that if
$ M\subset \mathbb{R}^3$, then $ K(X,Y)\ge 0$ for every $ Y\in T_{\id}\Diffmu(M)$ if and only if
$ X$ is a Killing field (i.e., one for which the flow generates a family of
isometries). This result was generalized by [Misio&lstrok;ek1993] and the second author
([Preston2002]) for any manifold with $ \dim{M}\ge 2$.
This gives the impression that, in general, $ D_\mu(M)$ will mostly be
negatively curved. The question of when one can expect a divergence free
vector field to give nonpositive sectional curvature remains open. However, the
second author ([Preston2005]) provided criteria for divergence free
vector fields of the form $ X = u(r)\partial_\theta$ on the area-preserving diffeomorphism
groups of a rotationally-symmetric surface for which the sectional curvature
$ K(X,Y)$ is nonpositive for all $ Y$.

Our goal in this paper is to extend the curvature computation to
$ \DiffmuE(M)$, the group of volumorphisms commuting with the flow of a
Killing field $ \Reeb$. In particular, we consider the solid flat torus,
$ M= D^2\times S^1$, where $ D^2$ is the unit disk in $ \mathbb{R}^2$ and
$ S^1$ is the unit circle, with cylindrical coordinates $ (r,\theta,z)$ for
$ 0\le r\le 1$ and $ \theta,z\in [0,2\pi]$. We may think of this more concretely as the
subset of $ \mathbb{R}^3$ with the planes $ z=0$ and $ z=2\pi$
identified, where $ E = \partial_\theta$ is the field corresponding to rotation in the disc.
Fluid flows on this manifold correspond to axisymmetric ideal flows with swirl
on the solid infinite cylinder, which are $ 2\pi$-periodic in the
$ z$-direction. We consider steady fluid velocity fields of the form
$ X = u(r)\partial_\theta$. The submanifold $ \DiffmuE(M)$ is a totally geodesic submanifold
of $ \Diffmu(M)$ (see [Vizman1999], as well as [Haller et al.2002, Modin et al.2011] for the general
situation in the smooth context, or see the preprint [Ebin and Preston2013] for the Sobolev
diffeomorphism context), corresponding to the fact that an ideal fluid which is
initially independent of $ \theta$ will always remain so. Hence we compute
sectional curvatures $ K(X,Y)$ where $ Y\in T_{\id}\DiffmuE(M)$ is divergence-free and
axisymmetric, i.e., $ [\Reeb,Y]=0$.

In [Preston2005] the second author effectively showed
that when $ X$ was considered as an element of $ \DiffmuF(M)$ where
$ F = \frac{\partial}{\partial z}$ (corresponding to considering $ X$ as a two-dimensional
flow rather than a three-dimensional flow), the sectional curvature satisfied
$ K(X,Y)\le 0$ for every $ Y\in T_{\id}\DiffmuF(M)$ regardless of $ u(r)$. By contrast we
show here that if $ u$ satisfies the condition

then $ K(X,Y)> 0$ for every $ Y\in T_{\id}\DiffmuE(M)$. We will also show that
$ \frac{d}{dr}\big( ru(r)^2\big) \geq 0$ implies that $ K(X,Y)\geq 0$. This does not contradict the result of
Rouchon, since the proof of that result relies on being able to construct a
divergence-free velocity field with small support which points in a given
direction and is orthogonal to another direction, and there are not enough
divergence-free vector fields in the axisymmetric case to accomplish this here.

The fact that the curvature is strictly positive in every section
containing $ X$ makes it natural to ask whether there are conjugate
points along every such corresponding geodesic. Unfortunately the Rauch
comparison theorem cannot be used here, since $ \inf_{Y\in T_{\id}\DiffmuE(M)} K(X,Y) = 0$ even if
(1)
holds. Nonetheless we can show that as long as

the geodesic formed by $ X = u(r)\partial_{\theta}$ has infinitely many
monoconjugate points. It is easy to see that condition
(1)
implies
(2)
. We do this by solving the Jacobi equation explicitly. As in [Ebin et al.2006], where the case $ u(r) \equiv 1$ was
considered, we can prove that these monoconjugate points have an
epiconjugate point as a limit point, so that the differential of the exponential
map is not even weakly Fredholm.

2.
The Formula for Curvature

We first compute the curvature of $ \DiffmuE(M)$ by expanding in a
Fourier series in $ z$. Here all our vector fields and functions are
smooth on the compact manifold $ M$, so that convergence of the
series will never be an issue, as in the original computations of [Arnold2014]. If desired one could do the same
computations in the Sobolev $ H^s$ context, with $ s>5/2$, and
treat the curvature operator as a continuous linear operator in $ H^s$,
as done by [Misio&lstrok;ek1993], but the final curvature
formula is the same in either case. Our method here is similar to that of the
second author in [Preston2005], where the computations were
two-dimensional.

Notice first of all that any smooth vector field $ Y$ which is
tangent to $ \DiffmuE(M)$ at the identity must be divergence-free and must
commute with $ E=\frac{\partial}{\partial \theta}$. Therefore we can write in the form

where $ f(0,z)=g(0,z) = 0$ and $ g(1,z)$ is constant in $ z$ (in
order to be well-defined on the axis of symmetry and to have $ Y$
tangent to the boundary $ r=1$). We think of the term $ -\frac{g_z}{r} \partial_r + \frac{g_r}{r} \partial_z$ as
an analogue of the skew-gradient in two dimensions. We may express
$ Y$ in a Fourier series in $ z$ as $ Y(r,z) = \sum_{n\in\mathbb{Z}} Y_n(r,z)$ where

The left side of this equation is a standard Bessel differential operator,
and so the solution formula
(7)
is essentially just the variation of parameters formula together with an
integration by parts since $ I_0$ and $ K_0$ solve the
corresponding homogeneous equation. Here we can simply verify the solution:
taking the derivative of $ q_n(r)$, we obtain

The projection $ P(\nabla_YX)$ is the most complicated part of the
curvature formula
(5)
since $ P(\nabla_XX)=0$ for steady flows $ X$. Hence Lemma 1 easily gives
the following expression for the curvature tensor.

Proposition 2.

Let $ M = D^2\times S^1$. Suppose that $ X\in T_{\id}\DiffmuE(M)$
is defined by $ X = u(r)\partial_\theta$, and let $ Y_n$ be of the form
(4)
. Then the curvature tensor $ R(Y_n,X)X$ is given for $ n\ne 0$ by

where $ q_n$ is the solution of the ODE
(10)
. For $ n=0$ we get $ R(Y_0,X)X=0$.

Proof.

We
compute using formula
(5)
. First note that $ \nabla_X X = -ru^2\partial_r$, which is the gradient of a function. Thus
$ P(\nabla_X X ) = 0$.

With the formula for the projection $ P(\nabla_{Y_n}X)$ from Lemma 1 in hand, we
will get \begin{eqnarray*} \nabla_X(P(\nabla_{Y_n}X)) = inug_n\left(u'+\frac{u}{r}\right)e^{inz} \partial_r- \frac{(q_n'+rf_nu)ue^{inz}}{r}\,\partial_\theta \end{eqnarray*} for any nonzero integer $ n$.
We also easily compute \begin{eqnarray*} \nabla_{[X,Y_n]}X = -in g_n(r)u(r)u'(r)e^{inz} \, \partial_r. \end{eqnarray*}

So, $ R$ will be given by
(12)
.
⬜

The sectional curvature can now be computed explicitly using Lemma
1 and
Proposition 2; the formula simplifies substantially due to
Bessel function identities.

Theorem 3.

On $ M=D^2\times S^1$ with $ X = u(r) \, \partial_{\theta}$ and
$ Y$ expressed as in
(3)
, the non-normalized sectional curvature is given by $ \overline{K}(X,Y) = \sum_{n\in \mathbb{Z}} \overline{K}(X,Y_n)$, where
$ Y_n$ is expressed as in
(4)
and

where $ \eta(r) = \frac{d}{dr} \big( ru(r)^2\big)$. By the definitions
(8)
of $ H_n$ and $ J_n$, we see that the second term in
(14)
is \begin{eqnarray*} 4\pi^2 \int_0^1 \big(\overline{H_n'(r)} J_n(r) - \overline{J_n'(r)} H_n(r)\big)\,dr. \end{eqnarray*}

From here we adapt the corresponding computation in [Preston2006]. Integrating by parts and using the
fact that $ J_n(r)\overline{H_n}(r)\to 0$ as $ r\to 0$ or $ r\to 1$, we get

3.
Solution of the Jacobi Equation

It is natural to ask whether the positive curvature guaranteed by the
theorem above ensures the existence of conjugate points along the
corresponding geodesic. This is not automatic since although the sectional
curvature is positive in all sections containing the geodesic's tangent vector, it
is not bounded below by any positive constant because of Remark 4; hence the
Rauch comparison theorem cannot be applied directly (and in any case would
need to be proved in the present formal context of weak metrics on Fr[U+00B4]echet
manifolds). In this section we answer this question affirmatively by solving the
Jacobi equation more or less explicitly along such a geodesic, and show that in
fact conjugate points occur rather frequently.

where $ P$ is the orthogonal projection onto divergence-free
vector fields. The first equation is the linearized flow equation, while the
second is the linearized Euler equation used in stability analysis.

Suppose $ a_{m,n}=a_{m,-n}=\tfrac{1}{2}$ for some $ (m,n)$ with $ n\ne 0$, and
that all other $ a$ are zero and that every $ b$ is zero, so
that $ h(t,r,z) = \cos{\left( \frac{nt}{\lambda_{mn}}\right)} \phi_{mn}(r) \cos{nz}$. Then by Eq.
(19)
we compute that \begin{eqnarray*} j(t,r,z) = -\frac{\lambda_{mn} \omega(r)}{r^2} \, \phi_{mn}(r) \sin{nz} \sin{\left( \frac{nt}{\lambda_{mn}}\right)}. \end{eqnarray*}

Remark 6.

Using the Sturm comparison theorem we can
estimate the spacing of the eigenvalues $ \lambda_{mn}$ and show that for fixed
$ m$ the sequence $ \lambda_{mn}/n$ has a finite limit as $ n\to\infty$.
Just as in [Ebin et al.2006], this must be an epiconjugate
point. Therefore the differential of the exponential map is not even weakly
Fredholm along any geodesic of this form (which is to say the differential of
the exponential map, extended to a linear map in the weak Riemannian
$ L^2$ topology, is not a Fredholm operator). It is worth noting that
the reason the Jacobi equation is explicitly solvable in this case is because
there is no "drift" term, so the total time derivative agrees with the partial
time derivative, in the same way as in [Ebin et al.2006].

It would be very interesting to generalize the curvature computation
to fields of the form $ X = u(r) \sin{z} \, \partial_{\theta}$, which is the initial velocity field of the
Luo-Hou initial condition ([Luo and Hou2014]) that leads numerically to a
blowup solution. We expect that the formula $ \int \overline{H_n'}J_n - \overline{J_n'}H_n$ which appears
both here and in [Preston2005] is a typical feature of curvature
formulas when computed correctly, although they doubtless become
substantially more complicated.

Acknowledgements.

The second author gratefully
acknowledges support from NSF Grants DMS-1157293 and DMS-1105660.

× Arnold, V.I.: On the differential
geometry of infinite-dimensional Lie groups and its application to the
hydrodynamics of perfect fluids. In: Arnold, V.I. (ed.) Collected works vol. 2.
Springer, New York (2014)